## Modelling Strategies for Spatial-Temporal Data

Spatial-temporal modelling has largely been developed through applications in geostatistics, hydrology and meteorology. More recent activities in the area include environment monitoring, tracking, functional MRI, health data and facial analysis. For a recent snapshot of activities, see Mardia et al. (1999). Motivated by these applications, the ﬁeld has adopted various modelling strategies. Current thinking in the ﬁeld has been surveyed by Haslett (1989), Goodall and Mardia (1994), Kyriakidis and Journel (1999), Mardia et al. (1998), Wikle and Cressie (1999) and Brown et al. (2000). The ideas behind these spatial-temporal models can be broadly cross-

classiﬁed according to (a) their motivation, (b) their underlying objectives and (c) the scale of data. Under (a) the motivations for models can be classiﬁed into four classes: (i) extensions of time series methods to space (ii) extension of random ﬁeld and imaging techniques to time (iii) interaction of time and space methods and (iv) physical models. Under (b) the main objectives can be viewed as either data reduction or prediction. Finally, under (c) the available data might be sparse or dense in time or space respectively, and the modelling approach often takes this scale of data into account. In addition, the data can be either continuously indexed or discretely indexed in space and/or time. Based on these considerations, especially (i) – (iii), we will describe several modelling strategies in detail and discuss their implementation. In Chapter 1 in this volume, space-time modelling concepts are introduced, while in Chapter 14, space-time modelling with a focus on object recognition is presented. For simplicity we assume that the data take the form of a regular array

in space-time. That is, we have one-dimensional observations

yij , i = 1, . . . , n, j = 1, . . . ,m, (12.1)

at sites xi ∈ Rd and at times tj ∈ R. Typically, d = 1, 2, or 3 for observations on the line, in the plane or in 3D space. However, we do not usually assume that the sites are regularly-spaced in Rd. The objective is to model the data as

yij = z(xi, tj) + ij , ij ∼ N(0, σ20), σ20 ≥ 0, (12.2) where {z(x, t), x ∈ Rd, t ∈ R} is a stochastic or deterministic space-time process. We usually assume the error terms ij are independent, identically distributed. In the language of geostatistics, such error terms are often known as a “nugget eﬀect”. Once a smooth process zˆ(x, t) has been ﬁtted, it can be used for interpolation and prediction. Both sites and times may be either continuously or discretely indexed according to context. Finally, we comment on notation. In general we use boldface to indi-

cate vectors, e.g. γ. In particular, sites in Rd will be denoted by x with components (x[1], . . . , x[d]).